fvade: Discretize scalar advection-diffusion equation using finite...

In Uffe-H-Thygesen/SDEtools: Utilities for Stochastic Differential Equations

Discretize scalar advection-diffusion equation using finite volumes

Description

fvade discretizes the advection-diffusion equation

dC/dt = -(u*C-D*C')'

on an interval [a,b] using the finite-volume method. A loss term can be included, as in

dC/dt = -(u*C-D*C')' - r*C

Usage

fvade(u, D, xgrid, bc, sparse = TRUE, diagonals = FALSE, r = NULL)

Arguments

u	function mapping state (numeric scalar) to advective term (numeric scalar)
D	function mapping state (numeric scalar) to diffusivity (numeric scalar)
xgrid	The numerical grid. Numeric vector of increasing values, giving cell boundaries
bc	String indicating boundary conditions. See details.
sparse	logical indicating if the result should be returned as a sparse matrix
diagonals	logical indicating if the result should be returned as a list of subdiagonal, diagonal, and superdiagonal
r	function mapping state (numeric scalar) to mortality/discount rate. Defaults to 0.

Details

Handling of boundary conditions: Input argument bc is a single character, or a vector of two characters, coding the condition at each boundary as follows: 'r': Reflecting boundary 'p': Periodic boundaries: Exit at this boundary to re-enter at the other 'a': Absorbing boundaries. In this case G will be a sub-generator 'c': Continue beyond boundary (experimental; read the source) 'e': Return generator, Extended to include absorbing boundaries

Return value: The function fvade returns a generator (or sub-generator) G of a continuous-time Markov Chain. This chain jumps between cells defined by xgrid. When using the generator to solve the Kolmogorov equations, note that G operates on probabilities of each cell, not on the probability density in each cell. The distinction is particularly important when the grid is non-uniform.

Value

a quadratic matrix, the (sub)generator of the approximating continuous-time Markov chain, with length(xgrid)-1 columns

Examples

# Generator of standard Brownian motion with unit drift on the unit interval, reflecting boundaries
xi <- seq(0,1,0.05)    # Cell boundaries
dx <- diff(xi)         # Cell widths
xc <- xi[-1] - 0.5*dx  # Cell centers
G <- fvade(function(x)1,function(x)0.5,seq(0,1,0.05),'r')

# Find the density of the stationary distribution
phi <- StationaryDistribution(G)         # Find stationary probabilities
phi <- phi/dx                            # Convert to densities
plot(xc,phi,type="l",xlab="x",ylab="Stationary density")

Uffe-H-Thygesen/SDEtools documentation built on Jan. 15, 2025, 6:41 p.m.